Download JEE MCQs for JEE Mathematics: Mathematical Reasoning
Explore reliable objective questions for Mathematical Reasoning tailored for JEE learners. Utilizing these Mathematics multiple-choice formats ensures thorough preparation and strengthens problem-solving speed for upcoming school assessments.
Chapter-wise Objective Questions: Mathematical Reasoning
Access the complete set of multiple-choice questions for Mathematical Reasoning below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.
Multiple Choice Questions
(a) May you live long!
(b) May God bless you!
(c) The sun is a star.
(d) Hurrah! we have won the match.
Show Answer & Explanation
Answer: (c) The sun is a star.
Explanation:
1. Wishes and exclamations are neither true nor false.
2. 'The sun is a star' is a true statement.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) 2 is an even integer.
(b) 2 + 1 = 3.
(c) The number 17 is prime.
(d) x + 3 = 10, x ∈ R
Show Answer & Explanation
Answer: (d) x + 3 = 10, x ∈ R
Explanation:
1. The first three are definitely true (or false).
2. 'x + 3 = 10' is an open sentence: its truth depends on x, so it is not a statement.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) and
(b) >
(c) or
(d) <
Show Answer & Explanation
Answer: (c) or
Explanation:
1. The two component statements are '2 + 7 > 9' and '2 + 7 < 9'.
2. They are joined by the word 'or'.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) or
(b) and
(c) Earth
(d) Sun
Show Answer & Explanation
Answer: (b) and
Explanation:
1. The components are 'Earth revolves round the Sun' and 'Moon is a satellite of earth'.
2. They are joined by 'and'.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) An ellipse is a circle.
(b) An ellipse is not a circle.
(c) A circle is an ellipse.
(d) A circle is not an ellipse.
Show Answer & Explanation
Answer: (d) A circle is not an ellipse.
Explanation:
1. The negation of a statement says the opposite about the same subject.
2. So it is 'A circle is not an ellipse'.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) Rajesh did not live in Bangalore or Rajni lived in Bangalore.
(b) Rajesh did not live in Bangalore and Rajni did not live in Bangalore.
(c) Rajesh lives in Bangalore and Rajni did not live in Bangalore.
(d) Rajesh did not live in Bangalore or Rajni did not live in Bangalore.
Show Answer & Explanation
Answer: (b) Rajesh did not live in Bangalore and Rajni did not live in Bangalore.
Explanation:
1. By De Morgan's law ~(p ∨ q) = ~p ∧ ~q.
2. So neither of them lived in Bangalore.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) If x² is odd, then x is even.
(b) If x is even, then x² is even.
(c) If x is not even, then x² is not even.
(d) If x is odd, then x² is even.
Show Answer & Explanation
Answer: (c) If x is not even, then x² is not even.
Explanation:
1. The converse of 'if p then q' is 'if q then p'.
2. The converse of 'if x is not even, then x² is not even' is 'if x² is not even, then x is not even'.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) p ∧ (~p) is a contradiction
(b) ~(~p) ⇔ p is a tautology
(c) p ∨ (~p) is a tautology
(d) (p ⇒ q) ⇔ (~q ⇒ ~p) is a contradiction
Show Answer & Explanation
Answer: (d) (p ⇒ q) ⇔ (~q ⇒ ~p) is a contradiction
Explanation:
1. p ∧ ~p is always false (contradiction) and p ∨ ~p is always true (tautology), and ~(~p) ⇔ p is always true.
2. A conditional and its contrapositive are equivalent, so (p ⇒ q) ⇔ (~q ⇒ ~p) is a tautology; calling it a contradiction is false.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) True
(b) False
(c) May be true or false
(d) Data insufficient
Show Answer & Explanation
Answer: (a) True
Explanation:
1. A conditional is false only when the antecedent is true and the consequent false.
2. q ∨ r is false since q = r = F; the antecedent p ∧ ~r must be true, and ~r = T, so p is true.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) p ∨ q
(b) p ∧ (~q)
(c) ~p ∧ ~q
(d) p ∨ (~q)
Show Answer & Explanation
Answer: (d) p ∨ (~q)
Explanation:
1. ~(A ∧ B) = ~A ∨ ~B.
2. So ~((~p) ∧ q) = p ∨ ~q.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) (~p ∨ ~q) → ~r
(b) (p ∨ q) → ~r
(c) (~p ∧ ~q) → ~r
(d) (~p ∧ q) → ~r
Show Answer & Explanation
Answer: (a) (~p ∨ ~q) → ~r
Explanation:
1. The inverse of A → B is ~A → ~B.
2. ~(p ∧ q) = ~p ∨ ~q, so the inverse is (~p ∨ ~q) → ~r.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) ~(p ∧ r) → ~(r ∨ q)
(b) (p ∧ r) ∧ (~r ∧ ~q)
(c) (~p ∨ ~r) ∨ (r ∨ q)
(d) (p ∧ r) ∨ (~r ∧ ~q)
Show Answer & Explanation
Answer: (b) (p ∧ r) ∧ (~r ∧ ~q)
Explanation:
1. ~(A → B) = A ∧ ~B.
2. Here ~(r ∨ q) = ~r ∧ ~q, so the negation is (p ∧ r) ∧ (~r ∧ ~q).
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) Where are you going?
(b) There are 35 days in a month.
(c) Please be quiet.
(d) Wow, what a match!
Show Answer & Explanation
Answer: (b) There are 35 days in a month.
Explanation:
1. There are 35 days in a month is false, but it is a definite statement.
2. The other three are a question, a request and an exclamation.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) √2 is a rational number.
(b) √2 is an irrational number.
(c) √2 is not a real number.
(d) √2 is a complex number.
Show Answer & Explanation
Answer: (d) √2 is a complex number.
Explanation:
1. The negation of 'not q' is q.
2. So the negation is '√2 is a complex number'.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) 'And', 'or'
(b) Today
(c) Yesterday
(d) Tomorrow
Show Answer & Explanation
Answer: (a) 'And', 'or'
Explanation:
1. Connectives join two simple statements into a compound statement.
2. Only 'and' and 'or' do this; the others are words about time.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) If I secure good marks, then I go for engineering.
(b) If I can go for engineering, then I secure good marks.
(c) If I go for engineering, then I do not secure good marks.
(d) If I cannot go for engineering, then I do not secure good marks.
Show Answer & Explanation
Answer: (b) If I can go for engineering, then I secure good marks.
Explanation:
1. Let p: I do not secure good marks, q: I cannot go for engineering. The contrapositive is ~q → ~p.
2. ~q: I can go for engineering and ~p: I secure good marks.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) F, T
(b) F, F
(c) T, T
(d) T, F
Show Answer & Explanation
Answer: (d) T, F
Explanation:
1. A conditional is false only when the antecedent is true and the consequent is false.
2. p = T and ~p ∨ q = F; since ~p = F, q must be F.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) ~p ∨ q
(b) p ∧ ~q
(c) ~q ⇒ p
(d) ~p ∧ q
Show Answer & Explanation
Answer: (a) ~p ∨ q
Explanation:
1. The truth table of p ⇒ q is false only for p = T, q = F.
2. ~p ∨ q has exactly the same truth table.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) false
(b) true
(c) true only if p is true
(d) cannot be determined
Show Answer & Explanation
Answer: (b) true
Explanation:
1. q → r = T → F = F.
2. p → (q → r) = F → F = T, since a false antecedent gives a true conditional.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) T, F, F
(b) F, F, F
(c) F, T, T
(d) T, T, F
Show Answer & Explanation
Answer: (d) T, T, F
Explanation:
1. The conditional is false only when p = T and (q ⇒ r) = F.
2. q ⇒ r is false only when q = T and r = F.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) p is false
(b) q is false
(c) p is true
(d) both p and q are false
Show Answer & Explanation
Answer: (d) both p and q are false
Explanation:
1. A disjunction is true if at least one component is true.
2. It is false only when both are false.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) p is true and q is false
(b) p is false but q is true
(c) at least one of p or q is false
(d) both p and q are false
Show Answer & Explanation
Answer: (a) p is true and q is false
Explanation:
1. Check the four rows of the truth table.
2. Only the row T, F gives F.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) p ⇔ q
(b) p ∧ q
(c) p ∨ q
(d) p ⇒ q
Show Answer & Explanation
Answer: (d) p ⇒ q
Explanation:
1. 'Implies' corresponds to a conditional.
2. So it is p ⇒ q.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) two
(b) one
(c) three
(d) four
Show Answer & Explanation
Answer: (a) two
Explanation:
1. Mathematical reasoning is of two types: inductive and deductive.
2. So the number is two.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) statement
(b) sentence only (not a statement)
(c) both a statement and an open sentence
(d) neither a statement nor a sentence
Show Answer & Explanation
Answer: (a) statement
Explanation:
1. It is a sentence that is definitely false.
2. A sentence that is either true or false but not both is a statement.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) Only Statement I is true
(b) Only Statement II is true
(c) Both statements are true
(d) Both statements are false
Show Answer & Explanation
Answer: (a) Only Statement I is true
Explanation:
1. I is correct.
2. The negation of 'every natural number is an integer' is 'there exists a natural number which is not an integer', so II is false.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
(a) I and II are the negation of each other
(b) I and II are not the negation of each other
(c) I and II are the contrapositive of each other
(d) I is the converse of II
Show Answer & Explanation
Answer: (b) I and II are not the negation of each other
Explanation:
1. Both statements are true.
2. Two statements that are both true cannot be negations of each other.
Teacher's Note:
1. Write the statement in symbols and use a truth table or the standard laws: p to q is equivalent to not p or q, and the negation of p to q is p and not q.
2. Remember: the converse swaps the parts, the contrapositive swaps and negates them, and the inverse only negates them.
Free study material for Mathematics
Multiple Choice Questions (MCQs) for JEE Mathematics Mathematical Reasoning
Download Multiple Choice Questions: Mathematical Reasoning (JEE Mathematics)
Explore reliable practice questions for Mathematical Reasoning tailored for JEE Mathematics learners. Use these multiple-choice formats to evaluate preparedness and strengthen problem-solving skills.
Concept Clarification for Mathematical Reasoning
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FAQs
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